Abstract

The impulsive complex-valued neural networks with three kinds of time delays including leakage delay, discrete delay, and distributed delay are considered. Based on the homeomorphism mapping principle of complex domain, a sufficient condition for the existence and uniqueness of the equilibrium point of the addressed complex-valued neural networks is proposed in terms of linear matrix inequality (LMI). By constructing appropriate Lyapunov-Krasovskii functionals, and employing the free weighting matrix method, several delay-dependent criteria for checking the global -stability of the complex-valued neural networks are established in LMIs. As direct applications of these results, several criteria on the exponential stability, power-stability, and log-stability are obtained. Two examples with simulations are provided to demonstrate the effectiveness of the proposed criteria.

1. Introduction

The nonlinear systems are ubiquitous in the real world [1–5]. As one of the most important nonlinear systems, the real-valued neural networks have been extensively studied and developed due to their extensive applications in pattern recognition, associative memory, signal processing, image processing, combinatorial optimization, and other areas [6]. In implementation of neural networks, however, time delays are unavoidably encountered [7]. It has been found that the existence of time delays may lead to instability and oscillation in a neural network [8]. Therefore, dynamics analysis of neural networks with time delays has received much attention. In [6–10], the exponential stability and asymptotic stability of delayed neural networks were investigated; some sufficient conditions for checking stability were given. In [11, 12], authors investigated the synchronization of chaotic neural networks with delay and obtained several criteria for checking the synchronization. In [13], the passivity of uncertain neural networks with both leakage delay and time-varying delay was considered. In [14], authors investigated the state estimation for neural networks with leakage delay and time-varying delays.

However, besides delay effect, impulsive effects are also likely to exist in neural networks [15]. For instance, in implementation of electronic networks, the state of the networks is subject to instantaneous perturbations and experiences abrupt change at certain instants, which may be caused by switching phenomenon, frequency change. or other sudden noise; that is, it exhibits impulsive effects [16–18]. Therefore, it is necessary to consider both impulsive effect and delay effect on dynamical behaviors of neural networks. Some results on impulsive effect have been gained for delayed neural networks; for example, see [15, 16, 19] and the references therein.

Recently, the power-rate global stability of the equilibrium is proposed in [20]. Moreover, in [21], the authors proposed a new concept of global -stability to unify the exponential stability, power-rate stability, and log-stability of neural networks. In [22], the authors investigated the global robust -stability in the mean square for a class of stochastic neural networks. In [23], the delayed neural systems with impulsion were considered, and the -stability criteria were derived by using Lyapunov-Krasovskii functional method. In [24], the multiple -stability of delayed neural networks was investigated, and several criteria for the coexistence of equilibrium points and their local -stability were derived.

As an extension of real-valued neural networks, complex-valued neural networks with complex-valued state, output, connection weight, and activation function become strongly desired because of their practical applications in physical systems dealing with electromagnetic, light, ultrasonic, and quantum waves [25, 26]. It has been shown that such applications strongly depend on the stability of CVNNs [27]. Therefore, stability analysis of CVNNs has received much attention and various stability conditions have been obtained [28–33]. In [28], authors considered a discrete-time CVNNs and obtained several sufficient conditions for checking global exponential stability of a unique equilibrium. In [29, 30], the discrete-time CVNNs with linear threshold neurons were investigated, and some conditions for the boundedness, global attractivity, and complete stability as well as global exponential stability of the considered neural networks were also derived. In [31], the continuous-time CVNNs with delays were considered, and the boundedness, complete stability, and exponential stability were investigated. In [32, 33], the global stability was investigated for CVNNs on time scales, which is useful to unify the continuous-time and discrete-time CVNNs under the same framework. In [34], the authors investigated the -stability of delayed CVNNs and obtained several sufficient conditions to ensure the global -stability. To the best of our knowledge, there are no results on -stability of impulsive CVNNs with the three kinds of time delays including leakage delay, discrete delay, and distributed delay in the literature, and it remains as an open topic for further investigation.

Motivated by the above discussions, in this paper, we will deal with the problem of -stability for CVNNs with leakage delay, discrete delay, and distributed delay under impulsive perturbations. Based on the homeomorphism mapping principle of complex domain, a LMI condition for the existence and uniqueness of the equilibrium point of the addressed CVNNs is proposed. Several delay-dependent criteria for checking the global -stability of the CVNNs are obtained by constructing appropriate Lyapunov-Krasovskii functionals and employing the free weighting matrix method. The obtained results can also be applied to several special cases and we can get the exponential stability, power-stability, and log-stability of the CVNNs, correspondingly. Finally, two illustrative examples are provided to show the effectiveness of the proposed criteria.

Notations. The notations are quite standard. Throughout this paper, let denote the set of positive integers. Let denote the imaginary unit; that is, . , , and denote, respectively, the set of -dimensional complex vectors, real matrices and complex matrices. The subscripts and denote matrix transposition and matrix conjugate transpose, respectively. For complex vector , let be the module of the vector , and let be the norm of the vector . denotes the identity matrix with appropriate dimensions. The notation (resp., ) means that is positive semidefinite (resp., positive definite). and are defined as the largest and the smallest eigenvalue of positive definite matrix , respectively. For any , is defined by , , . In addition, the notation always denotes the conjugate transpose of block in a Hermitian matrix.

2. Problems Formulation and Preliminaries

Consider the following complex-valued neural networks with leakage delay and mixed delays under impulsive perturbations by a nonlinear differential equation of the form
where the impulse times satisfy and ; = is the state vector of the neural networks; is the self-feedback connection weight matrix with (); , and are, respectively, the connection weight matrix, the discretely delayed connection weight matrix, and distributively delayed connection weight matrix; = represents the neuron activation function; is the external input vector; , , and are the leakage time delay, the discrete time delay, and the distributed time delay, respectively; is the impulsive function.

In the analysis of complex-valued neural networks, it is usually assumed that the activation functions are differentiable [31]. However, in this paper, we adopt the following assumption on the activation functions in which the differentiability is not be required:(H1)the neuron activation functions are continuous and satisfy
for any , , , where is a constant. Moreover, we define .

Remark 1. Note that the assumptions on activation functions are weaker than those generally used in the literature. Namely, the boundedness and differentiability of the activation functions are not required in this paper.

Next we introduce some definitions and lemmas to be used in the stability analysis.

Definition 2. Let be an equilibrium point of system (1). Suppose that is an arbitrary solution of system (1); is a positive continuous function and satisfies as . If there is a positive constant such that
then the equilibrium point is said to be -stable.

In particular, if taking in Definition 2 to exponential function, power function, and logarithmic function, we can get the definitions of exponential stability, power-stability, and log-stability, correspondingly.

Definition 3. Let be an equilibrium point of system (1). Suppose that is an arbitrary solution of system (1). If there are two positive constants and such that
then the equilibrium point is said to be exponentially stable.

Definition 4. Let be an equilibrium point of system (1). Suppose that is an arbitrary solution of system (1). If there are two positive constants and such that
then the equilibrium point is said to be power-stable.

Definition 5. Let be an equilibrium point of system (1). Suppose that is an arbitrary solution of system (1). If there are two positive constants and such that
then the equilibrium point is said to be log-stable.

Lemma 6. If is a continuous map and satisfies the following conditions: (i) is injective on ,(ii), then is a homeomorphism of onto itself.

Proof. Let and , where . Define a homeomorphism by
Obviously, is invertible. Let . Then is injective on , since and are injective. In addition, when , since and . Therefore, is a homeomorphism of onto itself. Then is a homeomorphism of onto itself. The proof is completed.

Lemma 7. For any , if is a positive definite Hermitian matrix, then .

Proof. Since is a positive definite Hermitian matrix, there exists an invertible matrix , such that . For any , it follows from Cauchy inequality that
The proof is completed.

Lemma 8 (see [33]). For any constant matrix and , a scalar function with scalars such that the integration concerned are well defined, then

Lemma 9. A given Hermitian matrix
where , and , is equivalent to any one of the following conditions: (1) and ,(2) and .

Proof. Note that
Therefore, and . Note that
Therefore, and . The proof is completed.

3. Existence and Uniqueness of Equilibrium Point

Now we study the existence and uniqueness of the equilibrium point of system (1). As usual, we denote an equilibrium point of the system (1) by the constant complex vector , where satisfies
In this paper, it is assumed that the constant complex vector satisfies for , if satisfies . Hence, to prove the existence and uniqueness of a solution of (13), it suffices to show that the following map has a unique zero point:
In the following, we will give some conditions for checking that is a homeomorphism on , that is, for assuring the existence and uniqueness of the equilibrium point of system (1).

Theorem 10. Under condition (H1), the system (1) has a unique equilibrium point, if there exist two complex matrices and a real positive diagonal matrix , such that the following LMI holds:
where , , + .

Proof. In the following, we will prove that is a homeomorphism of onto itself.First, we prove that is an injective map on . Suppose that there exist with , such that . Then
Multiply both sides above by ; we get
Taking the conjugate transpose of (17), we get that
Let , + − . Then is a positive definite Hermitian matrix from Lemma 9 and LMI (15). Summing (17) and (18), applying Lemma 7, we have
Since is a positive diagonal matrix, from condition (H1), we can get
It follows from (19) and (20) that
where . From Lemma 9 and LMI (15), we can know . Then from (21). Therefore, is an injective map on .Second, we prove as . Let . By Lemma 7, we have
It follows from condition (H1) that
Then from (22), (23), and , we obtain
which can imply that
When , we have
Therefore, as which implies as . From Lemma 6, we know that is a homeomorphism of . Thus, the system (1) has a unique equilibrium point.

Remark 11. It should be noted that Theorem 10 is independent of leakage time delay and initial conditions. So, the time delays in the leakage terms do not affect the existence and uniqueness of the equilibrium point.

4. Global -Stability Results

In the preceding section, we have shown the existence and uniqueness of the equilibrium point for system (1). In this section, we will further investigate the global -stability of the unique equilibrium point. For this purpose, the impulsive function which is viewed as a perturbation of the equilibrium point of system (1) without impulses is defined by
where , . It is obvious that .

Remark 12. The type of impulse such as (27) describes the fact that the instantaneous perturbations are not only related to the state of neurons at impulse times but also related to the state of neurons in recent history, which reflects more realistic dynamics [19].

Theorem 13. Under the conditions of Theorem 10, the equilibrium point of system (1) is globally -stable, if there exist six positive definite Hermitian matrices , , ,, , and , two real positive diagonal matrices and , three complex matrices , , and , a positive differential function , and three constants , , and such that
and the following LMIs hold:
where , , − , , , , and .

Proof. Under the condition of Theorem 10, system (1) has a unique equilibrium point . Then we shift the equilibrium point of (1) to the origin by the translation and obtain
Consider the following Lyapunov-Krasovskii functional candidate:
where
Calculating the upper right derivative of along the solution of (31), applying Lemma 8, we get
Combining (34), (35), (36), and (37), we can deduce that
From assumption (H1), we have
for . Let . It follows from (39) that
for . Hence
Also, we can get that
From (31), we have that
It follows from (38), (41), (42), and (43) that
where andwith , , , , , , . Thus, we get from (29) and (44) that
In addition, we note that
in which the last equivalent relation is obtained by Lemma 9. Thus, it yields
Hence, we can deduce that
From (46) and (49), we know that is monotonically nonincreasing for , which implies that
From the definition of in (32), we obtain that
where . It implies that
where . The proof is completed.

Corollary 14. Under the conditions of Theorem 10, the equilibrium point of system (1) is globally exponentially stable, if there exist six positive definite Hermitian matrices , , , , , and , two real positive diagonal matrices and , three complex matrices , , and , and a positive constant such that conditions (29) and (30) in Theorem 13 are satisfied, where , .

Proof. Let (); then , = . Take , , and . Then it is obvious that condition (28) in Theorem 13 is satisfied. The proof is completed.

Corollary 15. Under the conditions of Theorem 10, the equilibrium point of system (1) is globally power-stable, if there exist six positive definite Hermitian matrices , , , , , and , two real positive diagonal matrices and , three complex matrices , , and , and a positive constant such that conditions (29) and (30) in Theorem 13 are satisfied, where , .

Proof. Let . For all , we have , = . Take , , and . Then it is obvious that condition (28) in Theorem 13 is satisfied. The proof is completed.

Corollary 16. Under the conditions of Theorem 10, the equilibrium point of system (1) is globally log-stable, if there exist six positive definite Hermitian matrices , , ,, , and , two real positive diagonal matrices and , three complex matrices , , and , and a positive constant such that conditions (29) and (30) in Theorem 13 are satisfied, where , .

Proof. Let . For all , we have = , = ≥ . Take , , and . Then it is obvious that condition (28) in Theorem 13 is satisfied. The proof is completed.

Remark 17. It is noted that LMIs (15), (29), and (30) are complex-valued, which cannot be directly handled via MATLAB LMI Toolbox. However, the authors in [33] give the result that a complex Hermitian matrix satisfies if and only if
Therefore, applying the result, the complex-valued LMIs (15), (29), and (30) can be turned into real-valued LMIs, which can be checked numerically using LMI toolbox in MATLAB.

5. Numerical Examples

The following two illustrative examples will demonstrate the effectiveness and superiority of our results.

Example 1. Consider the following two-neuron CVNNs with leakage, discrete, and distributed delays, described by
where , , , , , , , and the parameter matrices , , , , and are given as follows:
It can be verified that the activation functions and satisfy condition (H1), and . Then LMI in Theorem 10 has the following feasible solution via the MATLAB LMI toolbox:
Also there exists a constant , and by employing MATLAB LMI Toolbox, we can find the solutions to the LMIs in Corollary 14 as follows: